I just want to remember myself that t-student compares two independent samples, so if we want to probe that averages from several samples are all equal, t-student must be carried out several times and that would change the critical value and the alpha value.
Say there are 4 (i =1,2,3 and 4) samples and their averages are µi. Our null hypothesis would be:
H0: µ1 = µ2 = µ3 = µ4
H1: averages are not all equal
t-student here would imply 6 comparisons (µ1 = µ2, µ1 = µ3, µ1 = µ4, µ2 = µ3,…) bringing a final alpha of 0.26 instead of the common 0.05
ANOVA analysis can do this. Three requirements are needed though:
Independence: k sample are independent,
Normality: all samples are distributed N(µi , σ2 i)
Homocedasticidad: all variance equal to σ2
If the F-value is less than the critical value (associated to alpha=0.05) and therefore P-value less than 0.05, then we reject the H0 (or better: we can not accept the H0).